Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

learn more… | top users | synonyms

4
votes
0answers
24 views

when a space is a product of eilenberg mclane spaces for $\pi_1(X)$ is not abelian

In When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$. , I discuss in my answer when an abelian cw complex $X$ is a product of Eilenberg-Maclane spaces, and show that it ...
1
vote
1answer
54 views

When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$.

Let $X$ be a connected CW complex. Let $X_n$ fit into a commutative postnikov diagram for $X$ and let the fibrations $K(\pi_n(X),n) \hookrightarrow X_n \xrightarrow{\mathscr p} X_{n-1}$ be given. ...
13
votes
2answers
253 views

Is wedge sum for finite CW complexes cancellative in the homotopy category?

Let $X,Y,Z$ be finite pointed CW complexes. Is it possible that $X\vee Z$ and $Y\vee Z$ are homotopy equivalent, but $X$ and $Y$ are not? Remark 1: Without the finiteness assumption on $Z$, there are ...
1
vote
1answer
46 views

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products?

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products? I know it preserves products, but it seems to actually create them.
8
votes
1answer
387 views

Why is the homotopy category actually important?

Since the homotopy category (of whatever) generally has very few limits and colimits, and these don't coincide with homotopy limits and colimits in the lifted model structure, why do we care about the ...
5
votes
2answers
87 views

What is wrong with this proof that the identity map of $S^1$ is nullhomotopic?

I have read that the identity map of the unit circle $S^1$ is not nullhomotopic. In fact, I am very new to the subject, so I wonder what is wrong with the following reasoning (that seems to suggest ...
2
votes
1answer
67 views

“Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
-2
votes
1answer
76 views

Is $(0,1)$ homotopic to $(0,1) \times (0,1)$? [on hold]

Is there a homotopy between a map with $(0,1) \in R^2$ as image and one with $(0,1) \times (0,1)\in R^2$ as image? Both have domain $(0,1)$. Or the same question for closed intervals.
1
vote
1answer
21 views

Homotopy 'diagrams' for Klein bottle and projective plane

Background: I recently discovered that the complement to the circle and vertical axis shown below is homotopy equivalent to a torus Also complement to three infinite straight non-intersecting ...
0
votes
1answer
30 views

Definition of G-crossed complex.

I was reading about crossed complexes following R.Brown. I was wondering how one define G-crossed complexes for a topological group G? Is it just dimension wise action of the group?
0
votes
0answers
35 views

How do I get these equations ($n$th-order deformation equation and its initial/boundary conditions)?

Earlier, I asked a question What does 'equating the like-power of $q$ ' mean? and I already got the answer about the meaning of a particular phrase in a certain context. However, my real question ...
4
votes
2answers
40 views

Homotopic retract vs deformation retract

Let's say that $A \subset X$ is a deformation retract. It follows that $A$ is both a retract and a space homotopically equivalent to $X$. Is the converse true? Probably not, but I couldn't find any ...
1
vote
1answer
26 views

What does 'equating the like-power of $q$' mean?

I am reading a book "Homotopy Analysis Method in Nonlinear Differential Equations" by Shijun Liao chapter 13 Applications in Finance: American Put Options. It is stated there that Substituting ...
0
votes
0answers
39 views

Dual of path object

For a topological space $X$, what might be the dual of the path space $X^I$ of $X$? Does it make any sense to think of the topological cylinder $X \times I$ over $X$ as dual to the path space over ...
3
votes
1answer
47 views

Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence.

Let $M,N$ be smooth manifolds. It seems to be well known that if the sets $C^0(M,N)$ and $C^\infty(M,N)$ are equipped with the appropriate topologies (I suppose the weak/strong Whitney topology), then ...
1
vote
2answers
61 views

A Natural Question When Reading Van Kampen Theorem

Let $A$ and $B$ be path connected open subspaces of a topological space $X$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x$ and $y$ be two points in $A\cap B$. Let $\gamma$ and ...
0
votes
2answers
100 views

Is torus w. disc removed homotopic to Klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know $f$ and $g$ are homotopic if they represent: ...
1
vote
1answer
18 views

principal bundle morphism preserves fundamental group

Two related questions. What is the morphism for principal bundles? Does it "preserve" fundamental groups? Fibre bundle morphisms usually preserve "the structure on the fibre". I am not sure how to ...
2
votes
1answer
48 views

Trying to Understand Van Kampen Theorem

Theorem. Let $X$ be the union of two path-connected open sets $A$ and $B$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x_0$ be a point in $A\cap B$ and all fundamental groups ...
0
votes
2answers
41 views

Is path-connected a homotopy property of toplogical spaces?

$X$ and $Y$ are homotopy equivalent so there are maps $\alpha: X \rightarrow Y$ and $\beta : Y \rightarrow X$ whose composites satisfy : $\beta\alpha \simeq id_X$ and $\alpha\beta \simeq id_Y$ $X$ is ...
0
votes
1answer
34 views

Are two equivariant maps between aspherical topological spaces homotopic?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial hight homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...
2
votes
0answers
40 views

Question about proving Cauchy's integral theorem

In my course we were given several proofs of Cauchy's theorem, each at various points in the course, each version stronger than the previous. I'd like to learn a proof of the theorem, so naturally ...
3
votes
1answer
60 views

Why does the homotopy lifting property imply that fibers are homotopy equivalent if the base is path connected?

Suppose that $\pi:E \to B$ has the homotopy lifting property, so that for any space Y with a map $f:Y \to E$ and a homotopy $G$ of $g = \pi \circ f$, we have a homotopy $F: Y \times I \to E$ that ...
1
vote
1answer
33 views

Fundamental group of the complement of $S^1 \cup Z \subset \mathbb{R^3}$

I want to calculate the fundamental group of the complement of this space: This is the complement of $S^1 \cup Z \subset \mathbb{R^3}$ where $S^1$ is the unit circle in the $xy$-plane and $Z$ is ...
3
votes
2answers
159 views

Homotopy Poincaré conjecture - no map inducing the isomorphism on homology

$\newcommand{\Z}{\mathbb{Z}}$ In Terence Tao's notes on page 18, concerning the Poincaré conjecture, he gave the following sketchy proof of the homotopy Poincaré conjecture. Given $M^3$ a 3-manifold ...
0
votes
0answers
12 views

Continuous maps to $S^n$ without antipodal pairs are homotopic [duplicate]

Let $S^n$ denote the unit sphere in the Euclidean space $\Bbb R^{n+1}$, $X$ a topological space, $f,g:X\to S^n$ are both continuous and there doesn't exist $x\in X$ such that $f(x)=-g(x)$, show ...
0
votes
0answers
24 views

Homotopy sets for a pushout of spaces (Seifert-van-Kampen?)

my problem is the following: I have two bordisms $M : \Sigma_0 \to \Sigma_1$ and $M' : \Sigma_1 \to \Sigma_2$, so I can glue them along $\Sigma_1$ to get $M'\circ M$. The manifold $M'\circ M$ is the ...
4
votes
1answer
480 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
2
votes
1answer
141 views

Motivation for the proof of the associativity of multiplication of equivalence classes of paths

After having defined the equivalence classes of paths in a topological space in chapter two of the book A Basic Course in Algebraic Topology, William S. Massey proves the lemma The multiplication ...
2
votes
2answers
79 views

Expressing homotopy groups of spaces of (unpointed) maps $S^1\to M$ in terms of homotopy groups of spaces of pointed maps.

I came across the following problem while studying for a topology exam: Let $M$ be a topological space, let $\Lambda(M)=M^{S^1}$, the space of continuous maps $S^1\to M$ with the compact-open ...
0
votes
1answer
49 views

Do homotopy equivalences operate over discrete spaces?

My understanding is limited and I'm trying to learn more about how homotopy forms the notion of equivalence. I can grasp its definition as "continuous", but my understanding of homotopy falls away in ...
0
votes
1answer
47 views

Homotopic type of $GL^+(n)$, $SL(n)$ and $SO(n)$

Question: Consider $GL^+(n) \supset SL(n) \supset SO(n)$ the groups of matrices $n \times n$ with positive determinant, determinant $1$ and orthogonal with positive determinant, respectively. Show ...
2
votes
2answers
48 views

Existence of a (n-1)-connected map beween CW-spaces

I have two finite CW-spaces $K$ and $L$ (K is n-dimensional and L is (n-1)-dimensional), a topological space $X$ and two maps $\phi:K\to X$ and $\psi: L\to X$, while $\phi$ is n-connected and $\psi$ ...
4
votes
0answers
155 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. It is true that if ...
1
vote
2answers
60 views

Does $f^{\ast}$ homotopic to $g^{\ast}$ imply $\int f^{\ast} w = \int g^{\ast} w$?

Let $f,g: M^{k} \to N$ ($M$ and $N$ with out boundary ) such that they are homotopic then for $\omega$ a $k$-form on $N$ do we have that $$ \int_M f^{\ast} \omega = \int_M g^{\ast} \omega$$ as ...
0
votes
0answers
15 views

Generalize equivalence of simply connected stated for subspaces of the complex plane

In complex theory we have the following proposition: Let $A \subset \mathbb{C}$ . Then A is simply connected (in the topological sense, i.e., that it's path connected and fundamental group is ...
2
votes
0answers
18 views

Homotopy continuations for solving systems of equations over a finite field

A way of solving systems of polynomial equations over $\mathbb{R}$ or $\mathbb{C}$ is using homotopy continuation. Roughly speaking this method uses a homotopy that starts from some system of ...
1
vote
1answer
64 views

Why does this have to be $f(0)=g(0)$?

For the problem, I am not given any solution so no idea Prove that any two continuous maps $f,g; I \to X$ such that $$f(0)=g(0) \in X$$ are homotopic where $I=[0,1]$ is the unit line. ...
3
votes
1answer
48 views

What do paths have anything to do with homotopy equivalence?

I don't understand how to solve this problem, it seems disconnected from the definition of homootpy equivalence Let $X,Y$ be spaces with the underyling set $\{a,b\}$ for both but ...
12
votes
3answers
130 views

Homotopy equivalence of a space with the sphere

I have some trouble with the following problem. A space $X$ is obtained by gluing two $2$-cells to a circle $S^1$ using maps winding $2$-times and $3$-times around $S^1$. Show that $X$ is homotopy ...
2
votes
1answer
24 views

Contradictory; Homotopy equivalence and deformation retract problem

The question Show that $S^1$ is a deformation retract og $D^2\setminus\{(0,0)\}$ the unit punctured disc. The solution the inclusion map $i:S^2 \to D^2\setminus\{(0,0)\}$ and ...
7
votes
0answers
156 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
0
votes
0answers
33 views

I don't understand what a Fundamental group is

I have been staring at the definition for days, drew diagrams but I don't understand as to what its elements are The fundamental group $\pi_1(X,x)$ at a base point $x$ is a set of rel $\{0,1\}$ ...
1
vote
2answers
14 views

Continuous deformation of loop to point.

Suppose I have a homotopy from a loop around the origin to a constant loop which is not the origin. Prove that the origin is in the image of the homotopy. Basically prove that if I deform a loop to ...
0
votes
1answer
35 views

“Reduction to finite case” arguments in algebraic topology

Hello I was studying the corollary to the excision property in Homotopy theory (Hatcher 4K.2) and the thing I can't understand is why the injectivity argument works when moving from an infinite ...
1
vote
0answers
26 views

Lifting paths in a fibration in families

Given a fibration (or a fiber bundle or whatever might make my question true) $f\colon E\rightarrow B$ and basepoints $e$ in $E$ and $b$ in $B$, such that $f$ preserves them. By the homotopy lifting ...
0
votes
1answer
22 views

Mind numbing homotopies; why is this a homotopy rel?

Proposition. every map $\alpha: [s_0,s_1] \to \mathbb{R}^n$ is homotopic rel $\{s_0,s_1\}$ to the linear map $$\beta=\frac{(s_1-s_0)\alpha(s_0)+(s-s_0)\alpha(s_1)}{s_1-s_0}:[s_0,s_1] \to ...
0
votes
2answers
34 views

$f$ is null homotopy if and only if $f_*$ is the trivial induced homomorphism

I want to show that $f:S^1\to S^1$ is homotopic to the constant map if and only if $f_*:\pi_1(S^1,x_0)\to \pi_1(S^1,x)$ , $f_*([\gamma])=0$ This seems like it should be an obvious fact but I am having ...
2
votes
0answers
30 views

Smoothing a continuous section in a fibre bundle.

Let $\pi: X \rightarrow M$ be a smooth fibre bundle and let $p^{1}_{0} : X^{(1)} \rightarrow X$ be its 1-jet bundle. Suppose there is a $\mathcal{C}^{1}$ section $h: M \rightarrow X$ such that it is ...
0
votes
1answer
49 views

Showing $\mathbb{R}^3$ minus $n$ parallel lines is homotopic to $\mathbb{R}^2\setminus\{p_1,\dots,p_n\}$

I want to show that if I remove $n$ parallel lines from $\mathbb{R}^3$ then I get $\mathbb{R}^2\setminus \{p_1,\dots,p_n\}.$ There is also some underlying structure I wish to also understand. That ...